2018/07/07 by Simon Henry, Henry, Simon
Mathematics · #18D05 #18G30 #18G55 #55U35 #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1807.02627
openalex publication_date 2018/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two different notions of "non-unital categories". The "general" version involve infinity-categories where absolutely all composition operations (horizontal, vertical and whiskering) are defined and compatible, the "regular" version involve infinity-categories where all the composition operations corresponding to "regular" pasting diagram are defined and compatible. In both case we construct (weak) model structures on these categories such that fibrant objects have weak units and weak inverse. We prove the regular version of the conjecture using the original strategy of Kapranov and Voevodsky, together with our previous work on polygraphs. The general version cannot be proved by these methods and is still open. In order to do this we also study some subtle property of the combinatorics of polygraphs, and we construct a new counting function for polygraphs, inspired by previous work of Makkai.